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Abstract

We address numerical dispersion compensation based on the use of the fractional Fourier transform (FrFT). The FrFT provides a new fundamental perspective on the nature and role of group-velocity dispersion in Fourier domain OCT. The dispersion induced by a 26 mm long water cell was compensated for a spectral bandwidth of 110 nm, allowing the theoretical axial resolution in air of 3.6 μm to be recovered from the dispersion degraded point spread function. Additionally, we present a new approach for depth dependent dispersion compensation based on numerical simulations. Finally, we show how the optimized fractional Fourier transform order parameter can be used to extract the group velocity dispersion coefficient of a material.

Figures (11)

Wigner distribution of a chirped signal in the time-frequency plane (u1, u). It illustrates how 2nd-order dispersion affects a spectral interferogram and how the continuum of domains of the FrFT can be interpreted geometrically by the rotated frame (ua, ua+1). The order parameter a chosen for the plot is optimal for the signal considered here, i.e., it leads to a dispersion-compensated OCT depth signal after FrFT.

(a) The dotted red curve is the PSF of our OCT system (3.8 μm FWHM) using only BK7 coarse physical dispersion compensation and the traditional FT (same as in Fig.3). The two other curves are obtained with an additional 26 mm long dispersive water cell. Using the traditional FT (solid blue curve), the PSF broadens to 49 μm FWHM, while the FrFT with optimized order aopt = 1.0555 numerically compensates dispersion, leading to a 3.65 μm FWHM PSF (dashed black curve). (b) Sharpness function as a function of fractional Fourier transform order, a.

(a) Spectral phase and (b) instantaneous spectral fringe frequency without (dotted red) and with the water cell (solid blue), and with the water cell plus FrFT dispersion compensation (dashed black). (c) and (d) shows the Wigner distribution of the spectrum acquired with the water cell before and after FrFT dispersion compensation.

OCT depth scan and intensity image of two cover slides (a) using the traditional FT and (b) using FrFT numerical dispersion compensation with the optimized order parameter. The 26 mm-long dispersive water cell is present in the setup in both cases.

(a) Radon transform of the spectrogram of one A-scan of a grape. (b) B-scan of the grape using the traditional FT (orange line). (c) B-scan of the grape using the optimized FrFT (aopt = 1.04, green line). The same FrFT order was used for all A-scans. The white bars correspond to 500 μm. Note that the x-axis of (a) only corresponds strictly to depth for the optimized FrFT so is labeled in terms of the pixels of the projections (or Radon bins).

(a) Sharpness function at each interface of a simulated sample made up five identical 100 μm-thick cover slides as a function of FrFT order. (b) Corresponding optimal FrFT order as a function of imaging depth.

Same as in Fig. 10 but for a simulated sample which presents linearly increasing normal dispersion in a first 150 μm-thick layer, followed by a second layer with uniform anomalous dispersion. (b) is obtained with the traditional FT (solid green line), while (c) comes from the optimized cross-section of the Radon plot (dashed yellow line).

Tables (1)

Table 1 FrFT-based measurements of the group velocity dispersion coefficient β2 of a single mode fiber and of distilled water. All the measurements have been done at 843 nm but the reference value for water is for a 840 nm wavelength. N = 1001, dω = 34.5×1010 rad/s.

Metrics

Table 1

FrFT-based measurements of the group velocity dispersion coefficient β2 of a single mode fiber and of distilled water. All the measurements have been done at 843 nm but the reference value for water is for a 840 nm wavelength. N = 1001, dω = 34.5×1010 rad/s.